A232172 Partial sums of second arithmetic derivative of natural numbers.

0, 0, 0, 4, 4, 5, 5, 21, 26, 27, 27, 59, 59, 65, 77, 157, 157, 167, 167, 211, 218, 219, 219, 267, 274, 282, 309, 389, 389, 390, 390, 566, 575, 576, 592, 684, 684, 694, 726, 798, 798, 799, 799, 911, 927, 937, 937, 1177, 1186, 1225, 1249, 1341, 1341, 1449, 1481

Offset: 1

Author

Giorgio Balzarotti, Nov 19 2013

Keywords

• nonn

Comments

a(n) = 1''+2''+3''+4''+5''+...+n'' -> ~ constant * n^2 as n -> oo.

Note: a(n) = sum(D^d(i)^m,i=1..n) -> constant * n^(m+1) as n -> oo where D^d(i) is the derivative of order d th of the natural number i (results on arithmetic derivatives descent from Barbeau's paper in References).

Examples

			a(5) = 1'' + 2'' + 3'' + 4'' + 5'' = 0+0+0+4+0 = 4.
		

Links

• E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull. vol. 4, no. 2, May 1961.

Crossrefs

Cf. A003415, A068346, A190121, A231946.

Programs

Maple

der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]): seq(add(der(der(i)),i=1..j),j=1..55);

Formula

a(n) = sum(i'', i=1..n), where i'' is the second arithmetic derivative of i (A068346).